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Mathematical models of NeolithisationMathematical models of Neolithisation
Joaquim Fort
Univ. de Girona (Catalonia, Spain)
FEPRE workshop26-27 March 2007
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List of ParticipantsList of Participants Kate Davison (Newcastle, UK) Pavel Dolukhanov (Newcastle, UK) Alexander Falileyev (Aberystwyth, UK) Sergei Fedotov (Manchester, UK) François Feugier (Newcastle, UK) Joaquim Fort (Girona, Spain) Neus Isern (Girona, Spain) Janusz Kozlowski (Krakow, Poland) Marc Vander Linden (Brussels, Belgium) David Moss (Manchester, UK) Joaquim Perez (Girona, Spain) Nicola Place (Newcastle, UK) Graeme Sarson (Newcastle, UK) Anvar Shukurov (Newcastle, UK) Ganna Zaitseva (St Petersburg, Russia)
FEPRE
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Diffusion Diffusion
time
4
DiffusionDiffusion
A A
J > 0 J < 0
tA t duringA area cross that particles ofnumber
Flux
J
5
J J = diffusion flux= diffusion flux
J < 0
J < 0
J = 0time
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J < 0
J = 0
c
xc
x
c = concentration = number particles / volume
0dx
dc
0dx
dc
J J = diffusion flux= diffusion flux
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Fick’s lawFick’s law
tcoefficiendiffusion
Ddxdc
DJ
c
x
c
x
0dx
dcDJ
0dx
dcDJ
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c
xc
x
c
x
How can we find out c(x,t) ?
time
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NN = = number of particles in volume number of particles in volume VV
Flux in 1 dimension:
J (x) J (x+x)V
JAxJxxJAAxxJAxJtN
)]()([)()(
dxdJ
VxdxdJ
A
A
dxdJ
dtVNd
dtdc )/(
xJ(x)
J(x+x) ∆ J
x ∆x
dxdJ
dtVNd
dtdc )/(
dxdJ
dtVNd
dtdc )/(
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How can we find out c(x,t) ?
dxdJ
dtdc
law sFick'
dxdc
DJ 2
2
dxcd
Ddtdc
We can find out c(x,t) !
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· Flux in 1 dimension:2
2
dx
cdD
dt
dc
If there is a chemical reaction:
2
2
2
2
dy
cd
dx
cdDF
dt
dc
· Flux in 2 dimensions:
2
2
2
2
dy
cd
dx
cdD
dt
dc
evolume·timproduced particles ofnumber F
For biological populations:
2
2
2
2
)(dy
pd
dx
pdDpF
dt
dp
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p0
pmax
p
time
a = initial growth rate
(of population number)
max
1)(p
ppapF
t
pLogistic growth:
?
atppdtap
pdpa
t
ppp
0max /ln
pmax= carrying capacity
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2 human populations:
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2
2
2
2
max
1y
p
x
pD
p
ppa
t
p
= jump distanceT = intergeneration dispersal time interval
Pre-industrial farmers (Majangir): < 2 > = (1544 ± 368 ) km2
T
D4
2
Fisher Eq:
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Dav 2
T
av2
km/yr4.1
yr25
km1544
yr032.022
1
v
T
a
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1.0 ± 0.2 km/yr observed
1.4 km/yr predicted by Fisher’s Eq. !!
10000 8000 6000 40000
1000
2000
3000
4000
5000
Ammerman & Cavalli-Sforza, 1971, 1984
r = 0.89
v = 1.0 km / yr
fit
dist
ance
( k
m )
date ( years B.P. )
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0 500 1000 1500 2000 2500 30000
1
2
3
4
< 2 > / T (km 2 /generation)
0.8
11.2
0.81
v = 1.2a (
%)
0.8 < v observed < 1.2 km/yr
Predictions from demic diffusion (Fisher's Eq.):
2 dimensions (F & M, PRL 1999)
1 dimension (A & C-S 1973)
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x
txcD
t
txJtxJ
),(),(),(
dx
txdcDtxJ
),(),( Up to now:
(Fick’s law)
Now:
→ instantaneous !
dx
txdcDtxJ
),(),(
→ time-delayed
(Maxwell-Cattaneo Eq.)dx
dfxxffxfxxf )()()(
f(x)
f(x+x)
Time delaysTime delays
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HRD EquationHRD Equation
dx
dcDJ
Fdx
cdD
dt
dc
2
2
Balance
of mass:
Now:
x
cD
t
JJ
Fdx
dJ
dt
dc
2
2
2
2
t
FF
x
cD
t
c
t
c
(HRD Eq.=Hyperbolic reaction-diffusion)
(Fisher’s Eq.)
Up to now:
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HRD Equation:HRD Equation:
For a
biological
population
in 2 dims:
2
2
2
2
t
FF
x
cD
t
c
t
c
2
2
2
2
2
2
t
FF
y
p
x
pD
t
p
t
p
max
1p
ppaFLogistic
reproduction:
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= jump (or migration) distance
T = time interval between the jumps of parents and those of their sons/daughters
T
D4
2
HRD Equation:
2
T
2
2
2
2
2
2
t
FF
y
p
x
pD
t
p
t
p
max
1p
ppaF
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Relationship with Fisher’s equationRelationship with Fisher’s equation
22 2
2
2
2
t
FTF
x
pD
t
p
t
pT
2 x
cD
t
JTJ
x
cDJ
Eq. HRD:
Fx
pD
t
p
2
2
(Fick’s law)
(Fisher’s Eq.)
<T > → 0
<T > → 0
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Dav 2
22
:Eq HRD 2
2
2
2
2
2
tFT
Fyp
xp
Dtp
tpT
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2 Eq. HRD
TaDa
v
max
1p
ppaF
<T > → 0(Fisher)
T
D4
2
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0 500 1000 1500 2000 2500 30000
1
2
3
4
0.8 < v obs
< 1.2 km / yr
< 2 > / T (km2 /generation)
time-delayed 0.8
1.0 km/yr
1.2
a (
%)
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SummarySummary
Observed Neolithic speed: 1.0 km/yr
Fisher’s equation in 2D: 1.4 km/yrHRD Eq: 1.0 km/yrDifference: 40 %
(F & M, Phys. Rev. Lett. 1999)
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Previous work by the Girona groupPrevious work by the Girona group
HRD Eq: F & M, Phys. Rev. Lett. 1999 ∞ terms: F & M, Phys. Rev. E 1999 Farmers + hunters: Phys. Rev. E 1999, Physica A 2006 Neolithic in Austronesia: F, Antiquity 2003 Several delays: Phys Rev E 2004, 2006 Paleolithic: F, P & Cavalli-Sforza, CAJ 2004 735 Neolithic sites: P, F & Ammerman, PLoS Biol 2006 Review: F & M, Rep. Progr. Phys. 2002 etc.